22 research outputs found
Embedding of global attractors and their dynamics
Using shape theory and the concept of cellularity, we show that if is the
global attractor associated with a dissipative partial differential equation in
a real Hilbert space and the set has finite Assouad dimension ,
then there is an ordinary differential equation in , with , that has unique solutions and reproduces the dynamics on . Moreover,
the dynamical system generated by this new ordinary differential equation has a
global attractor arbitrarily close to , where is a homeomorphism
from into
A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations
We put forward a new method for obtaining quantitative lower bounds on the
top Lyapunov exponent of stochastic differential equations (SDEs). Our method
combines (i) an (apparently new) identity connecting the top Lyapunov exponent
to a Fisher information-like functional of the stationary density of the Markov
process tracking tangent directions with (ii) a novel, quantitative version of
H\"ormander's hypoelliptic regularity theory in an framework which
estimates this (degenerate) Fisher information from below by an
W^{1,s}_{\loc} Sobolev norm. This method is applicable to a wide range of
systems beyond the reach of currently existing mathematically rigorous methods.
As an initial application, we prove the positivity of the top Lyapunov exponent
for a class of weakly-dissipative, weakly forced SDE; in this paper we prove
that this class includes the Lorenz 96 model in any dimension, provided the
additive stochastic driving is applied to any consecutive pair of modes